Boarding Permission is a feature in the game which allows a captain to decide who gets to board their ship.
With Guild checked, ANY guildmate can teleport aboard at any time.
When set to Public, any pirate looking for a ship to board may pop up on your deck. Anyone not in your crew that has boarded will have to be individually crewed to increase your Crew Bonus. By deselecting all of the options, no other pirate can board your ship other than those already on.
You can change the permissions at any time.
In Ship PvP, the captain can also boot off a crewman they want to remove. A captain can remove a member of their crew during normal sailing, but they currently cannot kick them off the ship.
A problem that exists is that a pirate can teleport to a friend or guildmate already aboard, even if the permissions are closed